Empirical Bayes estimation of the guarantee lifetime in a two-parameter exponential distribution

We study empirical Bayes estimation of the guarantee lifetime θθ in a two-parameter exponential distribution having a probability density p(x|θ,β)=(1/β)exp(-(x-θ)/β)I(x-θ)p(x|θ,β)=(1/β)exp(-(x-θ)/β)I(x-θ) with unknown scale parameter ββ. An empirical Bayes estimator ϕn* is proposed and its associated asymptotic optimality is studied. It is shown that ϕn* is asymptotically optimal in the sense that its regret converges to zero at a rate n-2r/(2r+1)n-2r/(2r+1), where n is the number of past observations available and r is a positive integer related to the prior distribution G.